A trivial realization of Markov property, application of Cauchy-Schwarz and recalling \(P^{m} f\)  is \((\lambda _{\beta ^*})^{m} \Vert f\Vert _{L(d_{\beta ^*})} \)  Lipshcitz (see e.g., [1]}) in respective order, reveals:
\(\nonumber |Cov_{P_x} [f(x_n),& f(x_{n+m})]|= |Cov_{P_x} [f(x_n),P^{m}f(x_n)]| \le \sqrt{Var_{P_x}[f(x_n)]} \sqrt{Var_{P_x}[P^{m}f(x_n)]} \\ & \nonumber \le (\lambda _{\beta ^*})^{m} \bigg [C(\beta ^*,\hat{\alpha }) + k(\beta ^*,\hat{\alpha }) \log P^{n} W(x) \bigg ] \Vert f\Vert _{L(d_{\beta ^*})}^2 \\ &  \le (\lambda _{\beta ^*})^{m} \bigg [C(\beta ^*,\hat{\alpha }) + k(\beta ^*,\hat{\alpha }) \log \big ( \eta W(x) + \frac{\hat{C}}{1-\eta } \big ) \bigg ] \Vert f\Vert _{L(d_{\beta ^*})}^2,\) 
